Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 285, pp. 1–23.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXPONENTIAL P -STABILITY OF STOCHASTIC ∇-DYNAMIC
EQUATIONS ON DISCONNECTED SETS
HUU DU NGUYEN, THANH DIEU NGUYEN, ANH TUAN LE
Abstract. The aim of this article is to consider the existence of solutions,
finiteness of moments, and exponential p-stability of stochastic ∇-dynamic
equations on an arbitrary closed subset of R, via Lyapunov functions. This
work can be considered as a unification and generalization of works dealing
with random difference and stochastic differential equations.
1. Introduction
The direct method has become the most widely used tool for studying the exponential stability of stochastic equations. For differential equations, we mention
the very interesting book by Khas’minskii [12] in which author uses the Lyapunov
functions to study stability. Foss and Konstantopoulos [8] presented an overview of
stochastic stability methods, mostly motivated by stochastic network applications.
Socha [24] considered the exponential p-stability of singularly perturbed stochastic
systems for the “slow” and “fast” components of the full-order system. Govindan
[9] proved the existence and uniqueness of a mild solution under two sets of hypotheses and considered the exponential second moment stability of the solution
process for stochastic semilinear functional differential equations in a Hilbert space.
We also refer to [15, 16] in which authors considered stochastic asymptotic stability
and boundedness for stochastic differential equations with respect to semimartingale via multiple Lyapunov functions. The long-time behavior of densities of the
solutions is studied in [20] by using Khas’minskii function. For random difference
systems, we can refer the reader to [18, 22, 23], for stability of nonlinear systems.
Recently, a method for the unified analysis of equations of motion in continuous
and discrete cases within the framework of the theory of time scales has drawn a
lot attention. For deterministic cases, in [4], author used the Lyapunov function of
quadratic form to study the stability of linear dynamic equations. Hoffacker and
Tisdell examined the stability and instability of the equilibrium point of nonlinear
dynamic equations [13]. Martynyuk presented systematically the stability theory
of dynamic equations in [17].
While the stability of deterministic dynamic equations on time scales has been
investigated for a long time, as far as we know, there is not much in mathematical
2010 Mathematics Subject Classification. 60H10, 34A40, 34D20, 39A13, 34N05.
Key words and phrases. Differential operator; dynamic equation; exponential stability;
Itô’s formula; Lyapunov function.
c
2015
Texas State University.
Submitted April 4, 2013. Published November 11, 2015.
1
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H. D. NGUYEN, T. D. NGUYEN, A. T. LE
EJDE-2015/285
literature for the stochastic case, and no work dealing with the stability of stochastic dynamic equations. Here, we mention some of the first attempts on this
direction. In [11], the authors developed the theory of Brownian motion. Sanyal
in his Ph. D. Dissertation [21] tries to define stochastic integral and stochastic dynamic equations on time scale with the positive graininess. Lungan and Lupulescu
in [14] consider random dynamical systems with random ∆-integral. Gravagne and
Robert deal with the bilateral Laplace transforms in [10]. The Doob-Meyer decomposition theorem and definition of stochastic ∇-integral with respect to square
integrable martingale on any arbitrary time scale also Itô’s formula are studied in
[6, 7]. Recently, Bohner et al [3] investigate stochastic dynamic equations on time
scale by considering an integral with respect to the restriction of a standard Wiener
process on time scale. However, this way can not be applied to define the stochastic
integral in general case since when one deals with a martingale defined on time scale
and we do not know whenever it can be extended to a regular martingale on R.
The aim of this article is to use Lyapunov functions to consider the existence,
finiteness of moments, and long term behavior of solutions for ∇-stochastic dynamic
equations on arbitrary closed subset of R. We study
d∇ X(t) = f (t, X(t− ))d∇ t + g(t, X(t− ))d∇ M (t)
X(a) = xa ∈ Rd ,
t ∈ Ta ,
where (Mt )t∈Ta is a R-valued square integrable martingale and f : Ta × R → R
and g : Ta × R → R are two Borel functions. We emphasis that martingale M is
defined only on Ta . This work can be considered as a unification and generalization
of works dealing with these areas of stochastic difference and differential equations.
In working on stochastic multi-dimensional dynamic equations with respect to
discontinuous martingale on time scales, it rises many difficulties, especially the
complicated calculations and they require some improvements. Besides, some estimates of stochastic calculus for continuous time are not automatically valid on
arbitrary time scale and we need to change them into a suitable form to obtain
similar results.
The organization of this paper is as follows. We introduce some basic notion
and definitions for time scale and for square integrable martingales in Section 2.
Section 3 deals with the existence and the finiteness of moments of solutions for
stochastic dynamic equations with respect to a square integrable martingale in
case the coefficients satisfy locally Lipschitz conditions. Section 4 is concerned
with necessary and sufficient conditions for the exponential p-stability of stochastic
dynamic equations.
2. Preliminaries
Let T be a closed subset of R, enclosed with the topology inherited from the
standard topology on R. Let σ(t) = inf{s ∈ T : s > t}, µ(t) = σ(t) − t and ρ(t) =
sup{s ∈ T : s < t}, ν(t) = t − ρ(t) (supplemented by sup ∅ = inf T, inf ∅ = sup T). A
point t ∈ T is said to be right-dense if σ(t) = t, right-scattered if σ(t) > t, left-dense
if ρ(t) = t, left-scattered if ρ(t) < t and isolated if t is simultaneously right-scattered
and left-scattered. The set k T is defined to be T if T does not have a right-scattered
minimum; otherwise it is T without this right-scattered minimum. Similarly, Tk
is defined to be T if T does not have a left-scattered maximum; otherwise it is
T without this left-scattered maximum. A function f defined on T is regulated if
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EXPONENTIAL P -STABILITY
3
there exist the left-sided limit at every left-dense point and right-sided limit at every
right-dense point. A regulated function is called ld-continuous if it is continuous
at every left-dense point. Similarly, one has the notion of rd-continuous. For every
a, b ∈ T, by [a, b], we mean the set {t ∈ T : a ≤ t ≤ b}. Denote Ta = {t ∈ T : t ≥ a}
and by R (resp. R+ ) the set of all rd-continuous and regressive (resp. positive
regressive) functions. For any function f defined on T, we write f ρ for the function
f ◦ ρ; i.e., ftρ = f (ρ(t)) for all t ∈k T and limσ(s)↑t f (s) by f (t− ) or ft− if this limit
exists. It is easy to see that if t is left-scattered then ft− = ftρ . Let
I = {t : t is left-scattered}.
Clearly, the set I of all left-scattered points of T is at most countable.
Throughout of this paper, we suppose that the time scale T has bounded graininess, that is ν∗ = sup{ν(t) : t ∈k T} < ∞.
Let A be an increasing right continuous function defined on T. We denote by
associated with A. For any µA
µA
∇ -measurable function
∇ the Lebesgue ∇-measure
Rt
f : T → R we write a fτ ∇Aτ for the integral of f with respect to the measures µA
∇
Rt
on (a, t]. It is seen that the function t → a fτ ∇Aτ is cadlag. It is continuous if A
Rt
Rt
is continuous. In case A(t) ≡ t we write simply a fτ ∇τ for a fτ ∇Aτ . For details,
we can refer to [5].
In general, there is no relation between the ∆-integral and ∇-integral. However,
in case the integrand f is regulated one has
Z b
Z b
f (τ− )∇τ =
f (τ )∆τ ∀a, b ∈ Tk ,
a
a
Indeed, by [5, Theorem 6.5],
Z b
Z
X
f (τ )∆τ =
f (τ )dτ +
f (s)µ(s)
a
[a;b)
a≤s<b
Z
=
f (τ− )dτ +
(a,b]
X
Z
b
f (s− )ν(s) =
f (τ− )∇τ.
a
a<s≤b
Therefore, if p ∈ R then the exponential function ep (t, t0 ), defined by [2, Definition
2.30, pp. 59], is solution of the initial value problem
y ∇ (t) = p(t− )y(t− ), y(t0 ) = 1, t > t0 .
(2.1)
Also if p ∈ R, ep (t, t0 ) is the solution of the equation
y ∇ (t) = −p(t− )y(t),
where p(t) =
−p(t)
1+µ(t)p(t) .
y(t0 ) = 1,
t > t0 ,
Later, we need the following lemma.
Lemma 2.1 ([2, 7]). Let u(t) be a regulated function and ua , α ∈ R+ . Then, the
inequality
Z t
u(t) ≤ ua + α
u(τ− )∇τ ∀t ∈ Ta
a
implies
u(t) ≤ ua eα (t, a)
∀t ∈ Ta .
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H. D. NGUYEN, T. D. NGUYEN, A. T. LE
EJDE-2015/285
Let (Ω, F, {Ft }t∈Ta , P) be a probability space with filtration {Ft }t∈Ta satisfying
the usual conditions (i.e., {Ft }t∈Ta is increasing and ∩{Fρ(s) : s ∈ T, s > t} = Ft
for all t ∈ Ta while Fa contains all P -null sets). Denote by M2 the set of the square
integrable Ft -martingales and by Mr2 the subspace of the space M2 consisting of
martingales with continuous characteristics. For any M ∈ M2 , set
X
ct = Mt −
M
(Ms − Mρ(s) ).
s∈(a,t]
ct is an Ft -martingale and M
ct = M
cρ(t) for any t ∈ T. Further,
It is clear that M
X
cit = hM it −
hM
(hM is − hM iρ(s) ).
(2.2)
s∈(a,t]
c ∈ Mr . In this case, M
c can be extended to a
Therefore, M ∈ Mr2 if and only if M
2
regular martingale defined on R.
Denote by B the class of Borel sets in R whose closure does not contain the
point 0. Let δ(t, A) be the number of jumps of the M on the (a, t] whose values
fall into the set A ∈ B. Since the sample functions of the martingale M are
cadlag, the process δ(t, A) is defined with probability 1 for all t ∈ Ta , A ∈ B.
We extend its definition over the whole Ω by setting δ(t, A) ≡ 0 if the sample
t → Mt (ω) is not cadlag. Clearly the process δ(t, A) is Ft -adapted and its sample
functions are nonnegative, monotonically nondecreasing, continuous from the right
b A) for M
ct by a similar way. Let
and take on integer values. We also define δ(t,
e
δ(t, A) = ]{s ∈ (a, t] : Ms − Mρ(s) ∈ A}. It is evident that
b A) + δ(t,
e A).
δ(t, A) = δ(t,
(2.3)
b ·) and δ(t,
e ·) are measures.
Further, for fixed t, δ(t, ·), δ(t,
b
e
The functions δ(t, A), δ(t, A) and δ(t, A), t ∈ Ta are Ft -regular submartingales for
fixed A. By Doob-Meyer decomposition, each process has a unique representation
of the form
δ(t, A) = ζ(t, A) + π(t, A),
b A) = ζ(t,
b A) + π
δ(t,
b(t, A),
e A) = ζ(t,
e A) + π
δ(t,
e(t, A),
where π(t, A), π
b(t, A) and π
e(t, A) are natural increasing integrable processes and
b
e
ζ(t, A), ζ(t, A), ζ(t, A) are martingales. We find a version of these processes such
that they are measures when t is fixed. By denoting
ctc = M
ct − M
ctd ,
M
where
ctd =
M
Z tZ
b
uζ(∇τ,
du),
a
R
we obtain
cit = hM
cc it + hM
cd it ,
hM
cd it =
hM
Z tZ
a
R
u2 π
b(∇τ, du).
(2.4)
Throughout this article, we suppose that hM it is absolutely continuous with respect
to Lebesgue measure µ∇ , i.e., there exists Ft -adapted progressively measurable
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EXPONENTIAL P -STABILITY
5
process Kt such that
t
Z
hM it =
Kτ ∇τ.
(2.5)
a
Further, for any T ∈ Ta ,
P{ sup |Kt | ≤ N } = 1,
(2.6)
a≤t≤T
where N is a constant (possibly depending on T ).
cc it and hM
cd it are is absolutely conThe relations (2.2), (2.4) imply that hM
tinuous with respect to µ∇ on T. Thus, there exists Ft -adapted, progressively
b tc and K
b td satisfying
measurable bounded process K
Z t
Z t
c
c
d
c
b
c
b τd ∇τ,
hM it =
Kτ ∇τ, hM it =
K
a
a
and the following relation holds
b tc + K
b td ≤ N } = 1.
P{ sup K
a≤t≤T
Moreover, it is easy to show that π
b(t, A) is absolutely continuous with respect to
µ∇ on T, that is, it can be expressed as
Z t
b A)∇τ,
π
b(t, A) =
Υ(τ,
(2.7)
a
b A). Since B is generated
with an Ft -adapted, progressively measurable process Υ(t,
b A) such that the
by a countable family of Borel sets, we can find a version of Υ(t,
b A) is measurable and for t fixed, Υ(t,
b ·) is a measure. Hence, from
map t → Υ(t,
(2.4) we see that
Z tZ
cd it =
b du))∇τ.
hM
u2 Υ(τ,
a
This means that
bd =
K
t
R
Z
b du)).
u2 Υ(t,
R
For the process π
e(t, A) we can write
X
π
e(t, A) =
E[1A (Ms − Mρ(s) )Fρ(s) ].
s∈(a,t]
Putting
( E[1
e A) =
Υ(t,
A (Mt −Mρ(t) )|Fρ(t) ]
ν(t)
0
if ν(t) > 0,
if ν(t) = 0
yields
Z
π
e(t, A) =
t
e A)∇τ.
Υ(τ,
Further, by the definition if ν(t) > 0 we have
Z
E[Mt − Mρ(t) Fρ(t) ]
e
= 0,
uΥ(t, du) =
ν(t)
R
and
E[(Mt − Mρ(t) )2 Fρ(t) ]
hM it − hM iρ(t)
u Υ(t, du) =
=
.
ν(t)
ν(t)
R
Z
2e
(2.8)
a
(2.9)
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H. D. NGUYEN, T. D. NGUYEN, A. T. LE
EJDE-2015/285
b A) + Υ(t,
e A). From (2.3) we see that
Let Υ(t, A) = Υ(t,
Z t
π(t, A) =
Υ(τ, A)∇τ.
a
Denote by Lloc
1 (TRa , R) the family of real valued, Ft -progressively measurable proT
cesses f (t) with a |f (τ )|∇τ < +∞ a.s. for every T > a and by L2 (Ta ; M ) the
RT
space of all real valued, Ft -predictable processes φ(t) satisfying E a φ2 (τ )∇hM iτ <
∞, for any T > a. Consider a d-tuple of semimartingales X(t) = (X1 (t), . . . , Xd (t))
defined by
Z
t
Z
fi (τ ) ∇τ +
Xi (t) = Xi (a) +
a
t
gi (τ )∇Mτ ,
a
where fi ∈ Lloc
1 (Ta , R) and gi ∈ L2 (Ta ; M ) for i = 1, d. For any twice differentiable
function V , put
AV (t, x)
=
d
X
∂V (t, x)
∂xi
i=1
(1 − 1I (t))fi (t) + V (t, x + f (t)ν(t)) − V (t, x) Φ(t)
Z
d
X
1 X ∂ 2 V (t, x)
∂V (t, x)
bc −
b du)
gi (t)gj (t)K
g
(t)
uΥ(t,
i
t
2 i,j ∂xi xj
∂xi
R
i=1
Z
+ (V t, x + f (t)ν(t) + g(t)u − V (t, x + f (t)ν(t)))Υ(t, du),
(2.10)
+
R
with f = (f1 , f2 , . . . , fd ); g = (g1 , g2 , . . . , gd ) and
(
0
if t is left-dense
Φ(t) =
1/ν(t) if t is left-scattered.
Let C 1,2 (Ta × Rd ; R) be the set of all functions V (t, x) defined on Ta × Rd , having
continuous ∇-derivative in t and continuous second derivative in x. Using the Itô’s
formula in [7] we see that for any V ∈ C 1,2 (Ta × Rd ; R+ )
Z t
V (t, X(t)) − V (a, X(a)) −
V ∇τ (τ, X(τ− )) + AV (τ, X(τ− )) ∇τ
(2.11)
a
is a locally integrable martingale, where V ∇t is partial ∇-derivative of V (t, x) in t.
3. Existence of solutions and finiteness of moments for stochastic
dynamic equations
Consider a ∇-stochastic dynamic equations on T of the form
d∇ X(t) = f (t, X(t− ))d∇ t + g(t, X(t− ))d∇ M (t)
X(a) = xa ∈ Rd ,
t ∈ Ta ,
(3.1)
where f : Ta × Rd → Rd and g : Ta × Rd → Rd are two Borel functions. Under
the global Lipschitz and linear growth rate conditions of the coefficients f, g, there
exists uniquely a solution for Cauchy problem (3.1) (see: [7]). We now consider the
case where the coefficients are locally Lipschitz.
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EXPONENTIAL P -STABILITY
7
Theorem 3.1. Suppose that for any k > 0 and T > a, there exists a constant
LT,k > 0 such that
kf (t, x) − f (t, y)k2 ∨ kg(t, x) − g(t, y)k2 ≤ LT,k kx − yk2 ,
(3.2)
for all x, y ∈ Rd with kxk ∨ kyk ≤ k and t ∈ [a, T ]. Further, there are positive
constants c = c(T ); b = b(T ) and a nonnegative function V ∈ C 1,2 ([a, T ] × Rd ; R+ )
satisfying
V ∇t (t, x) + AV (t, x) ≤ cV (t− , x) + b
∀(t, x) ∈ [a, T ] × Rd ,
(3.3)
and limx→∞ inf t∈[a,T ] V (t, x) = ∞. Then, (3.1) has a unique solution Xa,xa (t)
defined on Ta . In addition, if there exists a positive constant c1 = c1 (T ) such that
c1 kxkp ≤ V (t, x)
∀(t, x) ∈ [a, T ] × Rd ,
(3.4)
then
EkXa,xa (t)kp ≤
1
b
(V (a, xa ) + )ec (t, a)
c1
c
∀t ∈ [a, T ].
Proof. For each k ≥ k0 = [kxa k] + 1, define the truncation function
(
f (t, x)
if kxk ≤ k
fk (t, x) =
kx
f (t, kxk ) if kxk > k,
and gk (t, x) is defined by a similar way. The functions fk and gk satisfy the global
Lipschitz condition and the linear growth rate condition. Hence, by [7, Theorem
3.2] there exists a unique solution Xk (·) to the equation
d∇ X(t) = fk (t, X(t− ))d∇ t + gk (t, X(t− ))d∇ M (t)
X(a) = xa ∈ Rd , ∀t ∈ [a, T ].
(3.5)
Define the stopping time
θk = inf{t ∈ [a, T ] : |Xk (t)| ≥ k},
θk0 = a.
It is easy to see that θk is increasing and
Xk (t) = Xk+1 (t)
if a ≤ t ≤ θk .
(3.6)
Let θ∞ = limk→∞ θk and the process Xa,xa (t) = X(t), a ≤ t ≤ θ∞ be given by
X(t) = Xk (t),
θk−1 ≤ t < θk ,
k ≥ k0 .
Using (3.6) one gets X(t ∧ θk ) = Xk (t ∧ θk ). It follows from (3.5) that
Z t∧θk
Z t∧θk
X(t ∧ θk ) = xa +
fk (τ, X(τ− ))∇τ +
gk (τ, X(τ− ))∇Mτ
a
Z
t∧θk
Z
a
t∧θk
f (τ, X(τ− ))∇τ +
= xa +
a
g(τ, X(τ− ))∇Mτ ,
a
for any t ∈ [a, T ] and k ≥ 1. We show that limk→∞ θk = T a.s. Indeed, by (2.11)
it yields
Z t∧θk E[V (θk ∧ t, X(θk ∧ t))] = V (a, xa ) + E
V ∇τ (τ, X(τ− )) + AV (τ, X(τ− )) ∇τ
a
Z t
≤ V (a, xa ) +
(cEV (θk ∧ τ− , X(θk ∧ τ− )) + b)∇τ.
a
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H. D. NGUYEN, T. D. NGUYEN, A. T. LE
EJDE-2015/285
Using Lemma 2.1 with the function u(t) = E[V (θk ∧ t, X(θk ∧ t))] +
EV (θn ∧ t, X(θn ∧ t)) ≤ V (a, xa ) +
b
c
gets
b
ec (t, a).
c
On the other hand, on the set {θ∞ < T } we have lim supt→θ∞ kX(t)k = ∞. Therefore, the assumption limx→∞ inf t∈[a,T ] V (t, x) = ∞ follows P{θ∞ < T } = 0, i.e.,
the solution Xa,xa (t) is defined on Ta .
The uniqueness follows immediately from the uniqueness of solutions of (3.5).
When the condition (3.4) is satisfied we see that
b
c1 EkXa,xa (t ∧ θn )kp ≤ E[V (t ∧ θn , Xa,xa (t ∧ θn ))] ≤ (V (a, xa ) + )ec (t, a).
c
Letting n → ∞ yields
b
c1 EkXa,xa (t)kp ≤ (V (a, xa ) + )ec (t, a)
c
or
EkXa,xa (t)kp ≤
1
b
(V (a, xa ) + )ec (t, a).
c1
c
The proof is complete.
Corollary 3.2. Suppose that the conditions (2.5); (2.6) and (3.2) hold and the
linear growth condition
kf (t, x)k2 ∨ kg(t, x)k2 ≤ G(1 + kxk2 ) ∀(t, x) ∈ [a, T ] × Rd ,
(3.7)
R
b du) ≤ m1 a.s where m1 is a constant.
is satisfied. We suppose further that R |u|Υ(t,
Then (3.1) has a unique solution Xa,xa (t) defined on Ta satisfying
EkXa,xa (t)k2 ≤ (1 + kxa k2 )ec (t, a),
where c is a constant.
R
Proof. From (2.4), (2.7), it follows that R u2 Υ(t, du) < N for all t ∈ [a, T ]. Using
the Lyapunov function V (t, x) = 1 + kxk2 gets
AV (t, x)
b tc
= 2(1 − 1I (t))xT f (t, x) + kg(t, x)k2 K
2
2
Z
T
b du)
+ (kx + f (t, x)ν(t)k − kxk )Φ(t) − 2x g(t, x) uΥ(t,
R
Z
2
+ (kx + f (t, x)ν(t) + g(t, x)u − kx + f (t, x)ν(t)k2 )Υ(t, du)
R
Z
Z
T
T
T
e
b du)
= 2x f (t, x) + 2x g(t, x) uΥ(t, du) + 2ν(t)f (t, x) g(t, x) uΥ(t,
R
R
Z
b tc + kf (t, x)k2 ν(t) + kg(t, x)k2 u2 Υ(t, du)
+ kg(t, x)k2 K
R
≤ (1 + G(1 + 2N + 2m1 ν∗ + ν∗ ))(1 + kxk2 ) = cV (x),
where c = 1 + G(1 + 2N + 2m1 ν∗ + ν∗ ). Moreover, kxk2 ≤ 1 + kxk2 = V (x). Thus,
(3.3) and (3.4) are satisfied. Using Theorem 3.1 we can complete the proof.
EJDE-2015/285
EXPONENTIAL P -STABILITY
9
We note that in the continuous case, if the linear growth rate condition (3.7) holds
and the boundedness conditions (2.5), (2.6) of characteristic hM it are satisfied, then
all moments of the solutions are finite. This property may no longer valid on time
scale as it is shown in the following example.
Example 3.3. Consider two random variables ξ1 , ξ2 valued in Z \ {0} with
P{ξ1 = ±i} =
k
,
|i|5
1
P[ξ2 = j | ξ1 = i] = Ci |j|−(4+ |i| ) .
P
P
It is seen that 8/3 ≥ j∈Z\{0} |j|−4 > Ci−1 > j∈Z\{0} |j|−5 > 1 and E[ξ2 | ξ1 ] = 0.
Therefore, the sequence M1 = ξ1 and M2 = ξ1 + ξ2 is a martingale. Further,
X
E[ξ22 | ξ1 = i] =
j 2 P[ξ2 = j | ξ1 = i]
j∈Z\{0}
j2
X
= Ci
1
4+ |i|
|j|
j∈Z\{0}
X
≤ Ci
j∈Z\{0}
1
.
|j|2
Thus hM it is bounded. On the other hand,
X
E|ξ2 |3 =
|j|3 P[ξ2 = j | ξ1 = i]P{ξ1 = i}
i,j∈Z\{0}
X
=k
Ci
i,j∈Z\{0}
X
≤ 4k
i∈Z\{0}
1
1
(1+ |i|
)
|j|
|i|5
1
< ∞,
i4
which implies
E|M1 |3 < ∞, E|M2 |3 ≤ 4(Eξ13 + Eξ23 ) < ∞.
Consider the dynamic equation on the time scale T = {1, 2}
d∇ Xt = −Xt− d∇ t + Xt− d∇ Mt
X1 = ξ1 .
This equation has a unique solution X1 = ξ1 and X2 = ξ1 ξ2 . However,
X
E|X2 |3 = E|ξ1 ξ2 |3 =
|ij|3 P[ξ2 = j | ξ1 = i]P{ξ1 = i}
i,j∈Z\{0}
3k
≥
8
≥
3k
8
X
1
i,j∈Z\{0}
i2 |j|1+ |i|
1
X
i∈Z\{0}
|i|
1
= ∞.
i2
In the following we give conditions ensuring the finiteness of p-moment of the
solution of (3.1).
Theorem 3.4. Suppose that linear growth condition (3.7) and the conditions (2.5),
(2.6) hold. Further, there are two constants m1 , mp such that
Z
Z
b
|u|Υ(t, du) ≤ m1 ,
|u|p Υ(t, du) ≤ mp ∀t ∈ [a, T ]
(3.8)
R
R
10
H. D. NGUYEN, T. D. NGUYEN, A. T. LE
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almost surely. Then, the solution Xa,xa (t) of (3.1) starting in xa satisfies the
estimate
EkXa,xa (t)kp ≤ (kxa kp + 1)eH (t, a), a ≤ t ≤ T
(3.9)
where H is a constant.
R
Proof. Since R |u|2 Υ(t, du) = hM it ≤ N := m2 , we can suppose that p ≥ 2.
Applying (2.10) to the Lyapunov function V (t, x) = kxkp we have
AV (t, x) = pkxkp−2 (1 − 1I (t))xT f (t, x) + (kx + f (t, x)ν(t)kp − kxkp )Φ(t)
p
b c + p(p − 2) kxkp−4 |xT g(t, x)|2 K
bc
+ kxkp−2 kg(t, x)k2 K
t
t
2
2
Z
+ [kx + f (t, x)ν(t) + g(t, x)ukp − kx + f (t, x)ν(t)kp ]Υ(t, du)
R
Z
b du).
− pkxkp−2 xT g(t, x) uΥ(t,
R
Using Taylor’s expansion for the function kx + ykp at y = 0 obtains
kx + f (t, x)ν(t)kp − kxkp
p
= pkxkp−2 x> f (t, x)ν(t) + kx + θf (t, x)ν(t)kp−2 kf (t, x)k2 ν(t)2
2
p(p − 2)
+
kx + θf (t, x)ν(t)kp−4 |(x + θf (t, x)ν(t))> f (t, x)|2 ν(t)2 .
2
where 0 ≤ θ ≤ 1. It is seen that
kx + θf (t, x)ν(t)kp−2 kf (t, x)k2 ≤ (kxk + kf (t, x)ν(t)k)p−2 kf (t, x)k2
p
p
≤ ( 1 + kxk2 + G(1 + kxk2 )ν∗ )p−2 G(1 + kxk2 )
√
= G(1 + Gν∗ )p−2 (1 + kxk2 )p/2 ,
and
kx + θf (t, x)ν(t)kp−4 |(x + θf (t, x)ν(t))> f (t, x)|2
≤ kx + θf (t, x)ν(t)kp−2 kf (t, x)k2
√
≤ G(1 + Gν∗ )p−2 (1 + kxk2 )p/2 .
Similarly, the Taylor’s expansion of the function kx + f (t, x)ν(t) + ykp at y = 0
leads us
kx + f (t, x)ν(t) + g(t, x)ukp − kx + f (t, x)ν(t)kp
= pkx + f (t, x)ν(t)kp−2 (x + f (t, x)ν(t))> g(t, x)u
p
+ kx + f (t, x)ν(t) + ηg(t, x)ukp−2 kg(t, x)uk2
2
p(p − 2)
+
kx + f (t, x)ν(t) + ηg(t, x)ukp−4
2
× |(x + f (t, x)ν(t) + ηg(t, x)u)> g(t, x)u|2 ,
where 0 ≤ η ≤ 1. By defining cp = 2p−1 if p > 1 and cp = 1 if p ≤ 1 we have
kx + f (t, x)ν(t)kp−2 (x + f (t, x)ν(t))> g(t, x)u
≤ kx + f (t, x)ν(t)kp−1 kg(t, x)uk
√
√
≤ G(1 + Gν∗ )p−1 |u|(1 + kxk2 )p/2 ,
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EXPONENTIAL P -STABILITY
11
and
kx + f (t, x)ν(t) + ηg(t, x)ukp−2 kg(t, x)uk2
≤ cp−2 (kx + f (t, x)ν(t)kp−2 + (kg(t, x)ku)p−2 )kg(t, x)uk2
√
≤ cp−2 (G(1 + Gν∗ )p−2 u2 + Gp/2 |u|p )(1 + kxk2 )p/2 .
Further,
kx + f (t, x)ν(t) + ηg(t, x)ukp−4 |(x + f (t, x)ν(t) + ηg(t, x)u)> g(t, x)u|2
≤ kx + f (t, x)ν(t) + ηg(t, x)ukp−2 kg(t, x)uk2
√
≤ cp−2 (G(1 + Gν∗ )p−2 u2 + Gp/2 |u|p )(1 + kxk2 )p/2 .
Therefore, by using (2.9), (3.8) we obtain
AV (t, x)
= pkxkp−2 (1 − 1I (t))xT f (t, x)
p
+ (pkxkp−2 x> f (t, x)ν(t) + kx + θf (t, x)ν(t)kp−2 kf (t, x)k2 ν(t)2
2
p(p − 2)
+
kx + θf (t, x)ν(t)kp−4 |(x + θf (t, x)ν(t))> f (t, x)|2 ν(t)2 )Φ(t)
2
p
b tc + p(p − 2) kxkp−4 |xT g(t, x)|2 K
b tc
+ kxkp−2 kg(t, x)k2 K
2Z
2
e du)
kx + f (t, x)ν(t)kp−2 (x + f (t, x)ν(t))> g(t, x)uΥ(t,
+p
R
Z
+p
b du)
kx + f (t, x)ν(t)kp−2 (x + f (t, x)ν(t))> g(t, x)uΥ(t,
ZR
p
kx + f (t, x)ν(t) + ηg(t, x)ukp−2 kg(t, x)uk2 Υ(t, du)
+
2 R
Z
p(p − 2)
+
kx + f (t, x)ν(t) + ηg(t, x)ukp−4 |(x + f (t, x)ν(t)
2
R
Z
b du)
+ ηg(t, x)u)> g(t, x)u|2 Υ(t, du) − pkxkp−2 xT g(t, x) uΥ(t,
R
n √
√
p(p − 1)
≤ p G(1 + m1 ) +
G(N + (1 + Gν∗ )p−2 (ν∗ + cp−2 N ))
2
√
√
p(p − 1) p/2 o
p−1
+ p G(1 + Gν∗ ) m1 + cp−2
G mp (1 + kxk2 )p/2
2
≤ HV (x)
where H is defined
n √
√
p(p − 1) H = cp/2 p G(1 + m1 ) +
G N + (1 + Gν∗ )p−2 (ν∗ + cp−2 N )
2
(3.10)
√
√
p(p − 1) p/2 o
p−1
+ p G(1 + Gν∗ ) m1 + cp−2
G mp .
2
By Theorem 3.1, we obtain
EkXa,xa (t)kp ≤ (kxa kp + 1)eH (t, a),
The proof is complete.
a ≤ t ≤ T.
12
H. D. NGUYEN, T. D. NGUYEN, A. T. LE
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4. Exponential p-stability
By (2.1), the ∆-exponential function ep is also a solution of a ∇-dynamic equations. Therefore, in the following, instead of using ebp , we use ep to define the
exponential stability although we are working with stochastic ∇-dynamic equations. Let the process Kt be bounded on Ta , i.e., the constant N in (2.5) does
not depend on T > a. Suppose that for any s ≥ a; xs ∈ Rd , the solution Xs,xs (t)
with initial condition Xs,xs (s) = xs of (3.1) exists uniquely and it is defined on Ts .
Further,
f (t, 0) ≡ 0; g(t, 0) ≡ 0.
(4.1)
This assumption implies that (3.1) has the trivial solution Xs,0 (t) ≡ 0.
Definition 4.1. The trivial solution of (3.1) is said to be exponentially p-stable if
there is a positive constant α such that for any s > a there exists Γ = Γ(s) > 1,
such that
EkXs,xs (t)kp ≤ Γkxs kp eα (t, s) on t ≥ s,
(4.2)
d
holds for all xs ∈ R .
If one can choose Γ independent of s, the trivial solution of (3.1) is said to be
uniformly exponentially p-stable.
α
α
(t, s)
for all t ∈ T, 0 < eα (t, s) ≤ e− 1+αν
Remark 4.2. Since α(t) ≤ − 1+αν
∗
∗
α
and e− 1+αν
(t, s) → 0 as t → ∞. Thus, if α > 0 then limt→∞ eα (t, s) = 0. The
∗
advantage of using eα (t, s) is that the requirement −α ∈ R+ is not necessary.
Theorem 4.3. Suppose that there exist a function V (t, x) ∈ C 1,2 (Ta × Rd ; R+ ),
positive constants α1 , α2 , α3 such that
α1 kxkp ≤ V (t, x) ≤ α2 kxkp ,
V
∇t
(t, x) + AV (t, x) ≤ −α3 V (t− , x)
(4.3)
d
∀(t, x) ∈ Ta × R ,
(4.4)
where the differential operator Ais defined with respect to (3.1). Then, the trivial
solution x ≡ 0 of (3.1) is uniformly exponentially p-stable.
α
Proof. Let α be a positive number satisfying 1+αν(t)
< α3 for all t ∈ T and let
d
s ≥ a, xs ∈ R . To simplify notations, we write X(t) for Xs,xs (t). For each
n > kxs k, define the stopping time
θn = inf{t ≥ s :
kX(t)k ≥ n}.
Obviously, θn → ∞ as n → ∞ almost surely. By (4.13), calculating expectations
we obtain
E[eα (t ∧ θn , s)V (t ∧ θn , X(t ∧ θn ))]
Z t∧θn
h
= V (s, xs ) + E
eα (θn ∧ τ− , s) αV (τ− , X(τ− ))
s
i
+ (1 + αν(τ ))(V ∇τ (τ, X(τ− )) + AV (τ, X(τ− ))) ∇τ.
Using (4.4) and the inequality
α
1+αν(t)
< α3 obtains
αV (τ− , X(τ− )) + (1 + αν(τ )) V ∇τ (τ, X(τ− )) + AV (τ, X(τ− )) ≤ 0.
Therefore,
α1 eα (t ∧ θn , s)EkX(t ∧ θn )kp ≤ E[eα (t ∧ θn , s)V (t ∧ θn , X(t ∧ θn ))]
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EXPONENTIAL P -STABILITY
13
≤ V (s, xs ) ≤ α2 kxs kp .
Letting n → ∞ yields
α1 eα (t, s)EkX(t)kp ≤ α2 kxs kp .
Hence,
EkXs,xs (t)kp ≤
α2
kxs kp eα (t, s).
α1
The proof is complete.
We now consider the inverse problem by showing that if the trivial solution
of (3.1) is uniformly exponentially p-stable then such a Lyapunov function exits.
Firstly, we study the differentiability of solutions with respect to the initial conditions and the continuity with respect to coefficients.
Lemma 4.4 (Burkholder inequality on time scales). For any p ≥ 2 there exist
positive constants Bp such that if {Mt }t∈Ta is an Ft -martingale with E|Mt |p < ∞
and Ma = 0 then
X
p/2
E sup |Ms |p ≤ Bp EhM it + E
|∇∗ Ms |p ,
a≤s≤t
a≤s≤t
where ∇∗ Ms = Ms − Ms− .
Proof. By Doob’s inequality, we have
E sup |Ms |p ≤
a≤s≤t
p p
E|Mt |p .
p−1
ct can be extended to a regular martingale
Otherwise, we see that the martingale M
on [a; ∞)R . Therefore, by using proof of [19, Lemma 5] we obtain
X
∗c p
ct |p ≤ B
bp EhM
cip/2
E|M
+
E
|∇
M
|
,
s
t
a≤s≤t
bp . Further, the martingale M
ft is a sum of random variables. Then,
for a constant B
applying [1, Theorem 13.2.15, pp.416] yields
X
∗f p
ft |p ≤ B
ep EhM
fip/2
E|M
+
E
|∇
M
|
.
s
t
a≤s≤t
Consequently,
p p ct |p + E|M
ft |p
E|M
p−1
p p h X
∗c p
bp EhM
cip/2
e
f p/2
≤ 2p−1
B
+
E
|∇
M
|
+
B
s
p EhM it
t
p−1
a≤s≤t
i
X
fs |p
+E
|∇∗ M
E sup |Ms |p ≤ 2p−1
a≤s≤t
a≤s≤t
X
cit + hM
fit )p/2 + E
cs |p + |∇∗ M
fs |p )
≤ Bp E(hM
(|∇∗ M
a≤s≤t
X
p/2
= Bp EhM it + E
|∇∗ Ms |p
a≤s≤t
p p
bp , B
ep . The proof is complete.
where Bp = 2p ( p−1
) max B
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Theorem 4.5. Let p ≥ 2, M ∈ M2 such that the conditions (2.5), (2.6) and (3.8)
hold and let g ∈ L2 ((a, T ]; M ) with
Z t
E|g(τ )|p ∇τ < ∞ ∀t ∈ Ta .
a
Then
Z
p
Z t
g(τ )∇Mτ ≤ Cp
E sup a≤t≤T
where Cp = Bp {(T − a)
a
p
2 −1
N
T
E|g(τ )|p ∇τ,
a
p/2
+ mp }.
Proof. Set
t
Z
g(τ )∇Mτ ,
xt =
t ∈ [a, T ].
a
The process xt is a square martingale with the characteristic
Z t
hxit =
|g(τ )|2 ∇hM iτ .
a
Since hM it is continuous, so is hxit . Applying Lemma 4.4 to the martingale (xt )
obtains
E sup |xr |p
a≤r≤t
n
o
X
p/2
≤ Bp Ehxit + E
|∇∗ xs |p
a≤s≤t
Z tZ
n
o
p/2
= Bp Ehxit + E
|g(τ )u|p δ(∇τ, du)
a
R
Z tZ
n hZ t
ip/2
o
2
= Bp E
|g(τ )| ∇hM iτ
+E
|g(τ )u|p π(∇τ, du)
a
a
R
Z t
Z t
Z
n
o
p
−1 p/2
p
p
2
≤ Bp (t − a)
N
E|g(τ )| ∇τ + E
|g(τ )|
|u|p Υ(τ, du)∇τ
a
a
R
Z t
p
E|g(τ )|p ∇τ.
≤ Bp (t − a) 2 −1 N p/2 + mp
a
p
By putting Cp = Bp (T − a) 2 −1 N p/2 + mp we complete the proof.
Lemma 4.6. Let T, s ∈ Ta ; T > s and p ≥ 2 fixed. Suppose that the condition
(3.8) holds and process ζ(t) is the solution of the stochastic equation
Z t
Z t
ζ(t) = ϕ(t) +
ψ(τ )ζ(τ− )∇τ +
χ(τ )ζ(τ− )∇Mτ , ∀t ∈ [s, T ].
(4.5)
s
s
We assume that the functions ϕ(t), ψ(t) and χ(t) are Ft -adapted and that there
exists a constant K > 0 such that with probability 1, kψ(t)k ≤ K and kχ(t)k ≤ K.
Then
E sup kζ(t)kp ≤ 3p−1 E sup kϕ(t)kp eH1 (T, s),
(4.6)
s≤t≤T
p−1
where H1 = 3
p
K ((T − s)
s≤t≤T
p−1
+ Cp ).
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EXPONENTIAL P -STABILITY
15
Proof. For any n > 0 denote θn = inf{t > s : kζ(t)k > n}. From (4.5) we have
E sup kζ(r ∧ θn )kp
s≤r≤t
p−1
≤3
Z
E sup kϕ(r)k + E sup r∧θn
p
ψ(τ )ζ(τ− )∇τ p
s≤r≤T
Z
+ E sup s≤r≤t
s≤r≤t
r∧θn
s
s
p χ(τ )ζ(τ− )∇Mτ ≤ 3p−1 E sup kϕ(r)kp + K p (T − a)p−1
s≤r≤T
+ Cp K
p
Z
t∧θn
Z
Ekζ(τ− )kp ∇τ
s
t∧θn
Ekζ(τ− )kp ∇τ
(by Theorem 4.5)
s
p−1
=3
p
E sup kϕ(r)k + K
p
p−1
(T − a)
+ Cp
s≤r≤T
= 3p−1 E sup kϕ(r)kp + H1
s≤r≤T
Z
t∧θn
Ekζ(τ− )kp ∇τ
s
Z
t
sup Ekζ(r ∧ θn )kp ∇τ,
s s≤r≤τ−
where H1 = 3p−1 K p ((T − s)p−1 + Cp ). Using Lemma 2.1 one gets
E sup kζ(t ∧ θn )kp ≤ 3p−1 E sup kϕ(t)kp eH1 (T, s).
s≤t≤T
s≤t≤T
Letting n → ∞ yields (4.6). The proof is complete.
Lemma 4.7. Suppose that the coefficients of (3.1) are continuous in s, x and they
have continuous bounded first and second partial derivatives and condition (3.8)
holds for p ≥ 4. Then, the solution Xs,x (t), s ≤ t ≤ T , with initial condition
Xs,x (s) = x of (3.1) is twice differentiable with respect to x. Further, the derivatives
∂
(Xs,x (t)),
∂xi
are continuous in x in mean square.
∂2
(Xs,x (t))
∂xi ∂xj
00
00
Proof. Suppose that the derivatives fx0 (t, x), gx0 (t, x), fxx
(t, x), gxx
(t, x) are bounded
by a constant λ. To simplify notations we put Ys,∆x (t) = Xs,x+∆x (t) − Xs,x (t).
Using Lagrange theorem we see that for any i = 1, 2, . . . , d, there exists θi , ξi ∈ [0; 1]
such that
fi (t, Xs,x (t− ) + Ys,∆x (t− )) − fi (t, Xs,x (t− ))
=
d
X
∂fi
(t, Xs,x (t− ) + θi Ys,∆x (t− ))Yi,s,∆x (t− ),
∂xj
j=1
gi (t, Xs,x (t− ) + Ys,∆x (t− )) − gi (t, Xs,x (t− ))
=
(4.7)
d
X
∂gi
(t, Xs,x (t− ) + ξi Ys,∆x (t− ))Yi,s,∆x (t− ).
∂xj
j=1
Let As,∆x (t) be the matrix with entries aij
s,∆x (t) =
and let Bs,∆x (t) be the matrix with entries bij
s,∆x (t)
Then (4.7) can be rewritten
∂fi
∂xj (t, Xs,x (t− ) + θi Ys,∆x (t− )),
∂gi
= ∂x
(t, Xs,x (t− )+ξi Ys,∆x (t− )).
j
f (t, Xs,x (t− ) + Ys,∆x (t− )) − f (t, Xs,x (t− )) = As,∆x (t)Ys,∆x (t− ),
16
H. D. NGUYEN, T. D. NGUYEN, A. T. LE
EJDE-2015/285
g(t, Xs,x (t− ) + Ys,∆x (t− )) − g(t, Xs,x (t− )) = Bs,∆x (t)Ys,∆x (t− ).
Hence,
t
Z
Ys,∆x (t) = ∆x +
t
Z
As,∆x (τ )Ys,∆x (τ− )∇τ +
Bs,∆x (τ )Ys,∆x (τ− )∇Mτ .
s
s
Since As,∆x (t) and Bs,∆x (t) are bounded by a constant λ, by using Lemma 4.6 one
has
E sup kYs,∆x (t)k2 ≤ 3k∆xk2 eH2 (T, s),
(4.8)
s≤t≤T
2
where H2 = 3λ (T − s + C2 ). As a consequence, E sups≤t≤T kYs,∆x (t)k2 → 0 as
k∆xk → 0 in probability. Let ζs,x (t) be the solution of the variation dynamic
equation
Z t
Z t
0
fx (τ, Xs,x (τ− ))ζs,x (τ− )∇τ +
gx0 (τ, Xs,x (τ− )ζs,x (τ− )∇Mτ ,
ζs,x (t) = I +
s
s
fx0
for all s ≤ t ≤ T . Since
and
gx0
are bounded by constant λ,
E sup kζs,x (t)k4 ≤ 27eH3 (T, s),
(4.9)
s≤t≤T
where H3 = 27λ4 ((T − s)3 + C4 ). Define
ζ∆x (t) = Ys,∆x (t) − ζs,x (t)∆x ∀ s ≤ t ≤ T.
The process ζ∆x (t) satisfies Equation
Z t
Z t
ζ∆x (t) = φ∆x (t) +
As,∆x (τ )ζ∆x (τ− )∇τ +
Bs,∆x (τ )ζ∆x (τ− )∇Mτ ,
s
s
where,
Z
φ∆x (t) =
t
[(As,∆x (τ ) − fx0 (τ, Xs,x (τ− )))ζs,x (τ− )∆x]∇τ
s
Z t
+
[(Bs,∆x (τ ) − gx0 (τ, Xs,x (τ− )))ζs,x (τ− )∆x]∇Mτ .
s
Applying Lemma 4.6 again one gets
E sup kζ∆x (t)k2 ≤ 3E sup kφ∆x (t)k2 eH2 (T, s).
s≤t≤T
(4.10)
s≤t≤T
Since fx0 (t, x), gx0 (t, x) are continuous and E sups≤t≤T kYs,∆x (t)k2 → 0 as k∆xk → 0
in probability,
lim (kAs,∆x (t) − fx0 (t, Xs,x (t− ))k + kBs,∆x (t) − gx0 (t, Xs,x (t− ))k) = 0
∆x→0
in probability. Hence, by the boundedness of A, B, f 0 , g 0 , we obtain
kφ∆x (t)k2 E sup
k∆xk2
s≤t≤T
Z T
≤ 2(T − s)
EkAs,∆x (τ ) − fx0 (τ, Xs,x (τ− ))ζs,x (τ− )k2 ∇τ
s
Z
+8
s
T
EkBs,∆x (τ ) − gx0 (τ, Xs,x (τ− ))ζs,x (τ− )k2 ∇hM iτ → 0
(4.11)
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EXPONENTIAL P -STABILITY
17
as k∆xk → 0. Thus, (4.10) and (4.11) imply
E sup
t≤s≤T
kζ∆x (s)k
=0
k∆xk
as ∆x → 0.
This means
∂
Xs,x (t) ∀s ≤ t ≤ T.
∂x
The mean square continuity of ζs,x (t) with respect to x again follows from the
continuity of fx0 (t, Xs,x (t)) and gx0 (t, Xs,x (t)).
ζs,x (t) =
∂2X
(t)
s,x
We prove the existence of
. To simplify notations, if F is a bilinear
∂x2
2
mapping, we write F h for F (h, h). Let bilinear mapping ηs,x (t) be the solution of
the second variation dynamic equation
Z t
Z t
00
2
fx0 (τ, Xs,x (τ− ))ηs,x (τ− )∇τ
fxx (τ, Xs,x (τ− ))ζs,x (τ− )∇τ +
ηs,x (t) =
s
s
Z t
Z t
00
2
+
gxx (τ, Xs,x (τ− ))ζs,x
(τ− )∇Mτ +
gx0 (τ, Xs,x (τ− ))ηs,x (τ− )∇Mτ ,
s
s
for all s ≤ t ≤ T . Using Lemma 4.6 and (4.9) we see that
E sup kηs,x (t)k2 ≤ ∞.
(4.12)
s≤t≤T
Define
η∆x (t) = ζs,x+∆x (t)∆x − ζs,x (t)∆x − ηs,x (t)(∆x)2 ,
s ≤ t ≤ T.
The process η∆x (t) satisfies the equation
Z t
η∆x (t) = ψ∆x (t) +
fx0 (τ, Xs,x+∆x (τ− ))η∆x (τ− )∇τ
s
Z t
0
+
gx (τ, Xs,x+∆x (τ− ))η∆x (τ− )∇Mτ ,
(4.13)
s
where,
ψ∆x (t) =
Z t h
fx0 (τ, Xs,x+∆x (τ− )) − fx0 (τ, Xs,x (τ− ))
s
00
− fxx
(τ, Xs,x (τ− ) ζs,x (τ− )∆x ζs,x (τ− )∆x
i
+ (fx0 (τ, Xs,x+∆x (τ− )) − fx0 (τ, Xs,x (τ− )))ηs,x (τ− )(∆x)2 ∇τ
Z th
+
gx0 (τ, Xs,x+∆x (τ− )) − gx0 (τ, Xs,x (τ− ))
s
00
− gxx
(τ, Xs,x (τ− ))ζs,x (τ− )∆x ζs,x (τ− )∆x
i
+ gx0 (τ, Xs,x+∆x (τ− )) − gx0 (τ, Xs,x (τ− )) ηs,x (τ− )(∆x)2 ∇Mτ .
Using Lemma 4.6 one obtains
Ekη∆x (t)k2 ≤ E sup kψ∆x (t)k2 eH2 (T, s),
(4.14)
s≤t≤T
where H2 = 3λ2 (T − s + 4N ). It is easy to see that
Z
th 0
00
E sup fx (τ, Xs,x+∆x (τ− )) − fx0 (τ, Xs,x (τ− )) − fxx
(τ, Xs,x (τ− ))
s≤t≤T
s
18
H. D. NGUYEN, T. D. NGUYEN, A. T. LE
EJDE-2015/285
i 2
× ζs,x (τ− )∆x ζs,x (τ− )∆x ∇τ Z T
0
fx (τ, Xs,x+∆x (τ− ))
≤ 2(T − s)E
−
s
2
0
00
fx (τ, Xs,x (τ− )) − fxx
(τ, Xs,x (τ− ))Ys,∆x (τ− ) ζs,x (τ− )∆x ∇τ
Z T
00
fxx (τ, Xs,x (τ− ))(Ys,∆x (τ− ) − ζs,x (τ− )∆x)
+ 2(T − s)E
s
× ζs,x (τ− )∆xk2 ∇τ = o(k∆xk4 );
Z
T
s
2
E fx0 (τ, Xs,x+∆x (τ− )) − fx0 (τ, Xs,x (τ− )) ηs,x (τ− )(∆x)2 ∇τ
= o(k∆xk4 );
Z th
00
gx0 (τ, Xs,x+∆x (τ− )) − gx0 (τ, Xs,x (τ− )) − gxx
(τ, Xs,x (τ− ))
E sup s≤t≤T
s
2
i
× ζs,x (τ− )∆x ζs,x (τ− )∆x ∇Mτ Z T
0
gx (τ, Xs,x+∆x (τ− ))
≤ 4N E
−
+
s
2
0
00
gx (τ, Xs,x (τ− )) − gxx
(τ, Xs,x (τ− ))Ys,∆x (τ− ) ζs,x (τ− )∆x ∇τ
Z T
00
gxx (τ, Xs,x (τ− ))(Ys,∆x (τ− ) − ζs,x (τ− )∆x)ζs,x (τ− )∆xk2 ∇τ
4N E
s
4
= o(k∆xk );
Z t h
2
i
E sup gx0 (τ, Xs,x+∆x (τ− )) − gx0 (τ, Xs,x (τ− )) ηs,x (τ− )(∆x)2 ∇Mτ s≤t≤T
Z
s
T
≤ 4N E
0
(gx (τ, Xs,x+∆x (τ− )) − gx0 (τ, Xs,x (τ− )))ηs,x (τ− )(∆x)2 2 ∇τ
s
= o(k∆xk4 ).
Combining these results we obtain E sups≤t≤T kψ∆ (t)k2 = o(k∆xk4 ), which implies
that
Ekη∆x (t)k2 = o(k∆xk4 ).
Thus,
kη∆x (t)k
k∆xk2
= 0, or
∂2
Xs,x (t) = ηs,x (t).
∂x2
The proof is complete.
Lemma 4.8. Let p ≥ 4 and 2 ≤ β ≤ p. Then, the map F (φ) : φ → E|φ|β from
Lp (Ω, F, P) to R is twice differentiable at every φ0 6= 0 and
F 0 (φ0 )(φ) = βE[|φ0 |β−1 φ];
F 00 (φ0 )(φ, ψ) = β(β − 1)E[|φ0 |β−2 φψ].
Proof. We have
F (φ0 + ∆φ) − F (φ0 ) − βE|φ0 |β−1 ∆φ
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EXPONENTIAL P -STABILITY
19
= E|φ0 + ∆φ|β − E|φ0 |β − βE|φ0 |β−1 ∆φ
= β(β − 1)E[|η|β−2 (∆φ)2 ]
≤ β(β − 1)[E|η|m(β−2) ]1/m [E|∆φ|p ]2/p ,
where η ∈ (φ0 , φ0 + ∆φ) if φ0 + ∆φ > φ0 and η ∈ (φ0 + ∆φ, φ0 ) if φ0 + ∆φ < φ0 .
1
+ p2 = 1 we have
Hence, with m
F (φ0 + ∆φ) − F (φ0 ) − βE|φ0 |β−1 ∆φ
≤ β(β − 1)[E|η|m(β−2) ]1/m [E|∆φ|p ]2/p
≤ β(β − 1)[E max{|φ0 |, |φ0 + ∆φ|}m(β−2) ]1/m [E|∆φ|p ]2/p .
1
The relation m
+ p2 = 1 implies m(β −2) < p. Thus, E max{|φ0 |, |φ0 +∆φ|}m(β−2) <
∞. Therefore,
|F (φ0 + ∆φ) − F (φ0 ) − βE|φ0 |β−1 ∆φ|
≤ β(β − 1)[E|η|m(β−2) ]1/m [E(∆φ)p ]2/p
= O(1)|∆φ|2p
as |∆φ|p → 0.
This means F 0 (φ0 )(φ) = βE|φ0 |β−1 φ. The existence and continuity of the second
derivative F 00 con be proved by a similar way.
Lemma 4.9. Let the coefficients of (3.1) be continuous in t, x and satisfy the
conditions (4.1). Suppose also that conditions of Lemma 4.7 are satisfied and 2 ≤
β ≤ p. Then, for fixed t > a, the function u(s, x) = EkXs,x (t)kβ ; a < s < t is twice
continuously differentiable with respect to x except perhaps at x = 0.
Proof. The map x → Xs,x (t) is twice differentiable in x by Lemma 4.7. The map
X → kXk from Rd to R and the map F (φ) = E|φ|β from Lp (Ω, F, P) to R are
also twice differentiable. Therefore by chain rule, the map u(s, x) = EkXs,x (t)kβ is
twice differentiable. Further,
u0x (s, x)h = βE[kXs,x (t)kβ−2 < Xs,x (t), ζs,x (t)h >]
h
u00xx (s, x)h2 = βE (β − 2)kXs,x (t)kβ−4 hXs,x (t), ζs,x (t)hi2
(4.15)
i.
+ kXs,x (t)kβ−2 kζs,x (t)hk2 + kXs,x (t)kβ−2 hXs,x (t), ηs,x (t)h2 i
The proof is complete.
Theorem 4.10. Let M have independent increments and the conditions of Lemma
4.7 hold and 2 ≤ β ≤ p. Suppose further that AV (t, x) is ld-continuous in (t, x) for
all V ∈ C 1,2 (Ta × Rd ; R). Then, the function u(s, x) = EkXs,x (t)kβ , a < s < t is
∇-differentiable in s, twice continuously differentiable with respect to x and satisfies
the equation
u∇s (s, x) + Au(s, x) = 0.
(4.16)
Proof. By Lemma 4.7, u(s, x) is twice differentiable in x. From (4.2), (4.8), (4.9),
Rt
(4.12) and (4.15), it follows that s Au(h, Xs,x (τ− ))∇τ is integrable. Therefore,
Z r
u(h, Xρ(s),x (r)) − u(h, x) −
Au(h, Xρ(s),x (τ− ))∇τ, s ≤ r ≤ h ≤ t
ρ(s)
20
H. D. NGUYEN, T. D. NGUYEN, A. T. LE
EJDE-2015/285
is an Fr -martingale. In particular,
Z
h
Eu(h, Xρ(s),x (h)) − u(h, x) =
EAu(h, Xρ(s),x (τ− ))∇τ.
ρ(s)
Since Mt has independent increments, Xh,y (t) is independent of Xρ(s),x (h) when
s ≤ h ≤ t and y ∈ Rd , which implies that Eu(h, Xρ(s),x (h)) = u(ρ(s), x). Thus,
Z h
u(ρ(s), x) − u(h, x)
1
EAu(h, Xρ(s),x (τ− ))∇τ.
=
ρ(s) − h
ρ(s) − h ρ(s)
If s is left-scattered, then
u∇s (s, x) = −
1
ν(s)
Z
s
Au(s, Xρ(s),x (τ− ))∇τ = −Au(s, x).
ρ(s)
In the s is left-dense we let h → s to obtain
u∇s (s, x) = −Au(s, x).
The proof is complete.
Theorem 4.11. Let the conditions in Theorem 4.10 hold. Suppose that for any
fixed T > 0, there exist a function γT : T → T with γ(T, s) ≥ s + T for all
s ∈ T such that γ(T, s) and ∇-derivatives γ ∇s (T, s) are bounded. If the trivial
solution of (3.1) is uniformly exponentially β-stable, then there exists a function
V (s, x) ∈ C 1,2 (Ta × Rd ; R+ ) satisfying inequalities (4.3), (4.4) (with the power β).
Proof. By Lemma 4.9 and Theorem 4.10, the function
Z γ(T,s)
EkXs,x (τ− )kβ ∇τ,
V (s, x) =
(4.17)
s
is in class C 1,2 (Ta × Rd ; R+ ). From (4.2),
Z γ(T,s− )
V (s− , x) ≤
Γkxkβ eα (τ− , s− )∇τ ≤ α1 kxkβ ,
s−
Γ(1+ν∗ α)
.
α
where α1 =
By assumptions, the trivial solution of (3.1) is uniformly
exponentially β-stable and γ ∇s (T, s) is bounded, we can choose T > 0 such that
1
1
kxkβ , EkXs,x (γ(T, s))kβ γ ∇s (T, s) < kxkβ .
(4.18)
2
2
Since f and g have bounded partial derivatives and f (t, 0) = 0, g(t, 0) = 0,
EkXs,x (γ(T, s))kβ <
kf (t, x)k ≤ Gkxk,
kg(t, x)k ≤ Gkxk,
t ≥ a, x ∈ Rd .
Therefore,
kA[kxkβ ](s, x)k < c1 kxkβ ,
(4.19)
β
for a certain constant c1 . Applying Itô’s formula to the function kxk and using
(4.19) yields
Z γ(T,s)
EA(kXs,x (τ− )kβ )∇τ
EkXs,x (γ(T, s))kβ − kxkβ =
s
Z
≥ −c1
s
γ(T,s)
EkXs,x (τ− )kβ ∇τ = −c1 V (s, x).
EJDE-2015/285
EXPONENTIAL P -STABILITY
21
Combining with (4.18) we obtain the inequality V (s, x) > α2 kxkβ with α2 = 2c11 .
Thus, the function V satisfies condition (4.3). Using [2, Theorem 5.80] to calculate
∇-differential of V with respect s and applying Theorem 4.10 we obtain
V ∇s (s, x) + AV (s, x) = EkXs− ,x (γ(T, s− ))kβ γ ∇s (T, s− ) − kxkβ .
Using (4.18) again we have
1
1
V (s− , x).
V ∇s (s, x) + AV (s, x) ≤ − kxkβ ≤ −
2
2α1
Thus, the function V satisfies all conditions (4.3), (4.4) with α3 =
proof is complete.
α
2Γ(1+ν∗ α) .
The
Example 4.12. Consider the linear stochastic dynamic equation
d∇ X(t) = aX(t− )d∇ t + bX(t− )d∇ M (t) ∀t ∈ Ts
X(s) = x,
(4.20)
where a, b are two constants, a is regressive and M is a square integrable martingale
having independent increment. By direct calculation we have
Z t
2
2
EXs,x
(t) = x2 +
q(τ )EXs,x
(τ− )∇τ,
(4.21)
s
where
Z
c
2
b
q(t) = 2a + b Kt + a ν(t) + 2b(1 + aν(t)) uΥ(t, du)
R
Z
Z
2
2
b du)
+b
u Υ(t, du) − 2b uΥ(t,
R
R
Z
b tc + a2 ν(t) + 2b uΥ(t,
e du)
= 2a + b2 K
R
Z
Z
2
2
+b
u Υ(t, du) + aν(s) uΥ(t, du).
2
R
R
R
R
e du) = E[Mt − Mρ(t) |Fρ(t) ] = 0 and ν(t) uΥ(t, du) = 0,
Since R uΥ(t,
R
Z
b tc + a2 ν(t) + b2 u2 Υ(t, du).
q(t) = 2a + b2 K
(4.22)
R
We define the function q̄(t) = limρ(s)↓t q(s). It is seen that q̄ is rd-continuous
and q̄(t) = q(σ(t)) if t is right scattered. Since {t : µ(t) > 0} is countable and
2
2
meas{t : EXs,x
(t− ) 6= EXs,x
(t)} = 0,
Z t
Z
X
2
2
2
q(τ )EXs,x (τ− )∇τ =
q(τ )EXs,x
(τ− ) dτ +
q(τ )EXs,x
(τ− )ν(τ )
s
(s,t]
Z
s<τ ≤t
2
q̄(τ )EXs,x
(τ ) dτ +
=
[s,t)
Z
=
X
2
q(σ(τ ))EXs,x
(τ )µ(τ )
s≤τ <t
t
2
q(τ )EXs,x
(τ )∆τ,
s
from which it follows that
2
EXs,x
(t) = x2 eq (t, s),
t ≥ s.
(4.23)
22
H. D. NGUYEN, T. D. NGUYEN, A. T. LE
EJDE-2015/285
Further, it is known that
0 < eq (t, s) = exp
Z
t
lim
s h&µ(τ )
ln(1 + q(τ )h)
∆τ .
h
Then, condition (4.2) implies
Z t
ln(1 + q(τ )h)
lim
∆τ ≤ ln Γ − θ(t − s) ∀t > s.
h
h&µ(τ
)
s
Choose T > 0 such that ln Γ − θT
2 < 0 we obtain
Z t
ln(1 + q(τ )h)
θ(t − s)
lim
∆τ ≤ −
h
2
h&µ(τ
)
s
∀t > s + T.
Thus, the exponential square stability of (4.20) implies
Z t
1
ln(1 + q(τ )h)
lim
∆τ : t > s + T < 0.
sup
t − s s h&µ(τ )
h
(4.24)
Conversely, supposing that (4.24) holds, there are α > 0, K ∗ > 0 such that
0 < eq̄ (t, s) ≤ K ∗ e−α (t, s). By using (4.23) we see that the trivial solution of
is exponentially square stable. To illustrate the argument in the proof of Theorem
4.11 to construct a Lyapunov function we put
Z ∞
V (s, x) = x2
eq̄ (τ− , s)∇τ = x2 Q(s).
s
By direct calculation we have
Q∇s (s) = −1 − q(s)Q(s).
Hence,
V ∇s (s, x) + AV (s, x) = Q∇s (s)x2 + q(s)Q(s)x2
= (−q(s)Q(s) − 1)x2 + q(s)Q(s)x2 = −x2 .
(4.25)
2
(t) = 0 we can show that V (s, x) ≥ α1 x2
Using (4.21) and the fact limt→∞ EXs,x
∗
with α1 = (supt |q(t)|)−1 . Further, eq̄ (t, s) ≤ K ∗ e−α (t, s). Thus, V (s, x) ≤ Kα x2 .
Combining (4.25) and these inequalities obtains
α
V ∇s (s, x) + AV (s, x) ≤ − ∗ V (s− , x).
K
Thus, the conditions of Theorem 4.3 are satisfied. The proof is complete.
Acknowledgements. This research was supported in part by Vietnam National
Foundation for Science and Technology Development (NAFOSTED) No. 101.032014.58 also in part by Foundation for Science and Technology Development of
Vietnam’s Ministry of Education and Training. No. B2015-27-15.
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Huu Du Nguyen
Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334
Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
E-mail address: dunh@vnu.edu.vn
Thanh Dieu Nguyen
Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam
E-mail address: dieunguyen2008@gmail.com
Anh Tuan Le
Faculty of Fundamental Science, Hanoi University of Industry, Tu Liem district, Ha
Noi, Vietnam
E-mail address: tuansl83@yahoo.com